If you will look at Exodus 25:6, you will see that it does not say anything about oil for Holy Anointing Oil, it only talks about aromatics for it.
YLT Ex 25:6 oil for the light, spices for the anointing oil, and for the perfume of the spices,
Cross-referencing the word oil, produces a clue - Exodus 27:20
YLT Ex 27:20 'And thou--thou dost command the sons of Israel, and they bring unto thee pure beaten olive oil for the light, to cause the lamp to go up continually;
As you can see from the verse, it was [pure beaten olive] oil.
Cross-referencing the word beaten and pure produces the following verses:
YLT Ex 29:40 and a tenth deal of fine flour, mixed with beaten oil, a fourth part of a hin, and a libation, a fourth part of a hin, of wine, is for the one lamb.
YLT Num 28:5 and a tenth of the ephah of flour for a present, mixed with beaten oil, a fourth of the hin;
As you can see, the word "pure" is not present anywhere else, besides Exodus 27:20. Does this mean that "pure" oil was reserved for the lighting, and non-pure oil was acceptable for all other purposes?
Also, there is no word "olive" in Exodus 29:40 and Numbers 28:5. Does this mean it can be any oil? I.e vegetable oil?
I think the answer is more simple.
Since Exodus 25:6 does not mention any other type of oil, except the one for the lighting of the Menora lamps, we have to conclude that it was the only oil donated. Why use other, less quality oil for holy offerings and especially for anointing? This does not make sense. It does not say that any other type of oil was donated and it probably was not.
This means, that for all three purposes (offerings, lighting Menorah and anointing) they used "pure beaten olive oil" as Exodus 27:20 states.
This theory makes most sense to me in the absence of other evidence. Most likely, it is textual inconsistency and the original read "pure beaten olive oil" in all verses.